Basis Functions and Signal Representation - Signal Processing of Random Physiological Signals - page 23

4.4 SIGNAL REPRESENTATIONS
4.4.1 Fourier Transforms
The Trigonometric Fourier Transform Function. As indicated in the previous section,
it is convenient to use basis functions that are invariant with respect to the mathematical
operations, that is, integration, derivative, or summation. The “Sinusoidal Basis Func-
tion” has these properties and is commonly used to represent periodic, complex signals. It
is possible to use sines and cosines as basis functions as given by (4.15), but it is more con-
venient to use complex exponentials and to write the “Exponential Fourier Transform”
as (4.16).

∞

x ( t ) = a 0 + [ a n cos( n ω

0 t )

+ b n sin( n ω

0 t )]

(4.15)

n = 1

c n e − jn ω 0 t x ( t ) = (4.16)

You should be familiar with the relationship given in (4.17), and note that the
summation may be positive or negative.

e ± jnw 0 t = cos( nw 0 t ) ± j sin( nw 0 t ) (4.17)

Consider a function of time for which you would like to obtain the Fourier
series representation. The function is periodic over the interval, T t 1 .For

the series to converge to the true value, the Dirichlet conditions requires that the
function

= t 2 −

1. be single-valued within the interval, t 2 −

t 1 ;

2. have a finite number of maximums and minimums in a finite time;

3. satisfy the inequalities of (4.18)

t 2

x ( t ) dt

< ∞ ; and (4.18)

t 1

4. be a finite number of discontinuities.